Question

Let H be a subgroup of a group G. Prove that H is a normal subgroup of G if and only if, for all a and b in G, ab in H implies ba in H

Answer 1

what are you using for a definition of "normal"
sometimes that is used as one

Answer 2

A subgroup H of a group G is called a normal subgroup of G if aH=Ha for all a in G.

Answer 3

ok we can do this i think
we have to go both directions

Answer 4

probably easiest to start by assuming \(H\) is normal in \(G\) and showing that if \(ab\in H\) then \(ba\in H\)
since \(H\) is normal we have \(xHx^{-1}=H\) for all \(x\in G\)
if \(ab\in H\) then since for all \(x\in G, h\in H\) we have \(xhx^{-1}\in H\) and so
\[a^{-1}(ab)a\in H\] making \((a^{-1}a)ba\in H\) and so \(ba\in H\)

Answer 5

going the other way is very similar
if \(ab\in H\implies ba\in H\) then for any \(h\in H\) we have
\[h=(x^{-1}x)h=x^{-1}(xh)\in H\implies (xh)x^{-1}\in H\implies H\vartriangleleft G\]

Answer 6

thank you so much! This is great!

Answer 7

yw